~in the definition of the momentum operator. Our definitions ofSj and of
rotations using (see equation 6.3)
Ω(θ,w) =e−iθw·S=eθw·Xwill not include these factors of~, but in any case they will be equivalent to
the usual physics definitions when we make our standard choice of working with
units such that~= 1.
States inH=C^2 that have a well-defined value of the observableSjwill be
the eigenvectors ofSj, with value for the observable the corresponding eigen-
value, which will be±^12. Measurement theory postulates that if we perform the
measurement corresponding toSjon an arbitrary state|ψ〉, then we will
- with probabilityc+get a value of +^12 and leave the state in an eigenvector
|j,+^12 〉ofSjwith eigenvalue +^12 - with probabilityc−get a value of−^12 and leave the state in an eigenvector
|j,−^12 〉ofSjwith eigenvalue−^12
where if
|ψ〉=α|j,+1
2
〉+β|j,−1
2
〉
we have
c+=|α|^2
|α|^2 +|β|^2, c−=|β|^2
|α|^2 +|β|^2After such a measurement, any attempt to measure anotherSk,k 6 =jwill give
±^12 with equal probability (since the inner products of|j,±^12 〉and|k,±^12 〉are
equal up to a phase) and put the system in a corresponding eigenvector ofSk.
If a quantum system is in an arbitrary state|ψ〉it may not have a well-defined
value for some observableA, but the “expected value” ofAcan be calculated.
This is the sum over a basis ofHconsisting of eigenvectors (which will all
be orthogonal) of the corresponding eigenvalues, weighted by the probability
of their occurrence. The calculation of this sum in this case (A=Sj) using
expansion in eigenvectors ofSjgives
〈ψ|A|ψ〉
〈ψ|ψ〉=
(α〈j,+^12 |+β〈j,−^12 |)A(α|j,+^12 〉+β|j,−^12 〉)
(α〈j,+^12 |+β〈j,−^12 |)(α|j,+^12 〉+β|j,−^12 〉)=|α|^2 (+^12 ) +|β|^2 (−^12 )
|α|^2 +|β|^2=c+(+1
2
) +c−(−1
2
)
One often chooses to simplify such calculations by normalizing states so that
the denominator〈ψ|ψ〉is 1. Note that the same calculation works in general
for the probability of measuring the various eigenvalues of an observableA, as
long as one has orthogonality and completeness of eigenvectors.